Various native speakers have flagged that using "logics" is not correct because "logic" is uncountable. I have conducted some research and have found very few instances of plural "logics" being used. However, "logics" seems correct according to some grammar sources:
As with most uncountable nouns, logic may be plural, if it refers to different kinds of logic.
There certainly are different kinds of logic that have been written about and used.
Whether different kinds of logic have been applied in this case I don't know; but there is something else at play here: the whole sentence has a kind of rhetorical flourish, and can be read as saying something like "even if there is more than one kind of incontestable logic that has been applied, they are still not adequate to account for sociological analyses". In this reading, the mere possibility of there being more than one kind of logic makes the sentence acceptable.
There are surely circumstances in which a plural form "logics" can be correct. One may refer to multiple theories of logic, for example.
Science has used different logics at different times.
There are other senses of "logic" in which it is a countable noun.
That said, in most uses "logic" is not countable, and a plural form is not appropriate. It is hard to be sure without more context, but in the quoted sentence it seems that "*logic *" is being used as a non-countable noun, and so then form should be "logic", not "logics".